Confidence Interval Homework
Chapter 10
1. As marketing director of TXU Electric you are concerned about the average
household consumption of electricity each month. You take a sample of 49
households and determine a mean consumption of 1,834 KWH per month.
Previously you have determined that the standard deviation of consumption for all
households to be 175 KWH.
a. Calculate the standard error.
b. Calculate a 90% confidence interval.
c. For a sample of size 100 with the same mean calculate the standard error.
d. For a sample of size 100 with the same mean calculate a 90% confidence
interval. (Hey, with a larger sample size we can still be 90% confident
with a smaller interval!)
e. Considering the results of b., how big could our sampling error be such
that the true population mean was still in our estimated interval?
f. Considering the results of d., how big could our sampling error be such
that the true population mean was still in our estimated interval? (Hey
again, with a larger sample size the standard error is smaller which leads
to a smaller sampling error! With a smaller standard error, we are less
likely to suffer sampling error…the true population mean is just as likely
to be in our smaller interval!)
g. Explain two interpretations of the 90% confidence interval calculated in b.
2.
You are the proprietor of a NYC boutique. You want to know the average age of
your customers. You take a random sample of 25 customers which yields an
average age of 32 with a standard deviation of 8. You have reason to believe the
distribution of ages is normally distributed. Determine a 95% confidence interval
for the age of your customers.
3. A random sample of 61 children with working mothers showed that the children
were absent an average of 5.3 days per term with a standard deviation of 1.8 days.
Provide a 99% confidence interval for the average number of days absent per term
for all the children. Explain the meaning of the confidence interval.
4. The monthly starting salaries of students who receive an MBA degree have a
standard deviation of $70. What sample size should be selected so that there is a
90% confidence of estimating the mean monthly income within a sampling error
of $15 or less?
5. To estimate the mean expenditure of customers at a local Sonic, students in a
marketing research class sample 200 customers and find a mean expenditure of
$5.67. The standard deviation for all meal expenditures is believed to be $1.10.
Develop a 96% confidence interval for the mean expenditure for all customers.
Interpret the results.
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6. Jose has a thriving business in Acapulco selling authentic plastic Inca relics to
tourists. He selects n = 60 days to estimate his true daily profit. The sample
mean daily profit Jose obtains is $1248, with standard deviation $100. Jose
believes his daily profits are not normally distributed as he has a few extremely
profitable days each year that tend to skew the profit distribution. Jose wants to
obtain an interval for his true daily mean profit with extreme confidence (99%).
7. A survey is planned to determine the mean annual family medical expenses of
employees of a large company. The management of the company wishes to be
95% confident that the sample mean is correct to within +/- $50 of the true
population mean of annual family medical expenses. A pilot study indicated the
population standard deviation can be estimated to be $400.
a. How large a sample size is needed?
b. If management wants to be correct to within +/- $25, what sample size is
necessary?
8. A political pollster wants to determine a range for the true proportion of voters
who will vote for the Independent candidate in a presidential campaign. The
pollster wants to have 96% confidence that her range is correct.
She polls a sample of 100 registered voters and determines that 16 voters are
planning to vote for the Independent candidate.
a. Based on the above information, construct a confidence interval that will
provide a range for the proportion of voters who favor the Independent
candidate.
b. If the pollster wanted to be 99% confident in her estimate and have a
sampling error to within +/- .04 (or 4%), how big a sample size would she
need to take (use p=.16, our educated guess of p’s value)? Obtain the 99%
confidence interval.
Textbook 12.66, 12.68
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Homework
Confidence Intervals,  not known/given
1. a. The sample mean and standard deviation from a sample of 81 observations are
x-bar=63 and s=8. Estimate  with 95% confidence.
b. Repeat part a with n=64.
c. Repeat part a with n=36.
d. Describe what happens to the CI estimate when the sample size decreases.
2. The Bank Accounts Excel spreadsheet is a random sample of 40 student customer
balances. Derive a 90% CI for the population mean student balance. Balances
are believed to be normally distributed.
Helpful Excel formula
=average
=stdev
=tinv(2-tail total percentage, degrees of freedom)  returns the t-value
Answers
1. a. [61.23, 64.77]
b. [61, 65]
c. [60.29, 65.71]
d. the interval widens
2. [734, 962]
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Homework
Hypothesis Testing
1. Shell Oil claims that the average gasoline price of their brand is less than $1.25
per gallon. In order to test their claim, we randomly selected a sample of 49 of
their gas stations and determined that the average price per gallon of the stations
in the sample is $1.20. Furthermore, we assume that the standard deviation of the
population is $.14 from other studies. Using an alpha of .05, test the company’s
claim. State a conclusion that would be understandable to anyone.
Next, find the p-value and explain its meaning.
2. The manager of a local grocery store believes that the store’s average daily sales
are more than $8,000 per day. To test his belief, a sample of 64 days of sales was
selected and it was found that the average sale was $8,250 with a sample standard
deviation of $1,200. Test the manager’s belief using an alpha of .01. State a
conclusion that would be understandable to anyone.
Next, find the p-value and explain its meaning.
3. A lathe is set to cut bars of steel with a length of 6 centimeters. The lathe is
considered to be in perfect adjustment if the average length of the bars it cuts is 6
centimeters. A sample of 25 bars is selected randomly, and their lengths are
measured. It is determined that the average length of the bars in the sample is 6.1
centimeters with a standard deviation of .2 centimeters. At an alpha value of .05,
test to see if the lathe is in perfect adjustment. State a conclusion that would be
understandable to anyone.
Next, find the p-value and explain its meaning.
At an alpha value of .01, what would our conclusion be and why?
4. More professional women than ever before are foregoing motherhood because of
the time constraints of their careers. Yet, many women still manage to find time
to climb the corporate ladder and set time aside to have children. A survey of 187
attendees at Fortune Magazine’s Most Powerful Women in Business summit in
March 2002 found that 133 had at least one child. Assume the group of 187
women is a random sample from the population of all successful women
executives.
a. What is the sample proportion of successful women executives who have
children?
b. At the .05 level of significance, can you state that more than half of all
successful women executives have children?
c. At the .05 level of significance, can you state that more than two-thirds of
all successful women executives have children?
d. Do you think the random sample assumption is valid? Explain.
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Homework
Hypothesis Testing Using P-Value’s
1.
In the summer of 2001, Congress approved a federal budget containing several
provisions for tax reductions. The claim was that it would save the average
taxpayer $800. A sample of 500 taxpayers showed a mean reduction in taxes of
$785.10 with a standard deviation of $277.70. Test the hypothesis at the 5% alpha
level using the p-value. Interpret the p-value and comment on its meaning.
2.
Ben and Jerry sell ice cream from a pushcart in NYC Central Park. Ben tells
Jerry they sell an average of at least 15 pounds of vanilla when the temperature
exceeds 80 degrees. If 20 days of at least 80 degree weather reveals an average of
13.9 pounds with a standard deviation of 2.3 pounds, who is correct at the alpha
level of .05 using the p-value. Interpret the p-value and comment on its meaning.
3. Use Excel to get P-values using the data file MOISTURE. Boston and Vermont
make asphalt shingles. Builders (the customers for the shingles) may feel they
have purchased a product lacking in quality if they find moisture (measured in
pounds per hundred square feet) and wet shingles inside the packaging. Excess
moisture can cause appearance and construction problems. The manufacturers
monitor the amount of moisture present by taking samples of the singles received.
The file MOISTURE contains moisture levels for a sample from each
manufacturer. Each manufacturer would like to show that the moisture levels
for their product is less than .35.
Use the t methodology even though the sample sizes are slightly greater than 30.
a. What are the Null and Alternative Hypothesis?
b. Use Excel to determine the p-value for each sample.
c. What conclusions can you make?
Use Excel to complete the following steps:
 Find the mean of each sample.
 Find the standard deviation of each sample.
 Find n and the degrees of freedom for each sample.
 Calculate the test statistic t value for each sample.
 Find the P-Value for each sample. Use the Excel function
TDIST(ABS(t-test value), d.f., 1 tailed test)
 What conclusions would you make at the .05 LOS?
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Hypothesis Testing – Marketing Survey Example
An advertising firm has a client that is interested in buying an expensive
television commercial for air during the 2007 super bowl. The client will buy the
commercial if the ad firm can provide strong evidence that their commercial
provokes a sense of excitement in the viewer.
The commercial has been filmed and shown to two groups of likely super bowl
viewers. After viewing the commercial, each person completed a questionnaire
asking them about their feelings of excitement regarding the product that was
shown in the commercial. Feelings of excitement was measured on a 1 to 10
point scale with 10 being very excited about the product and 0 being not excited
at all. The ratings for each person in each group are contained in the
spreadsheet files labeled SampleA and SampleB.
The ad firm claims their ad provokes an excitement level rating of more than 6 on
a 10-point scale. If this claim can be substantiated, the client will proceed with
the purchase.
1. Perform two hypothesis tests (for groups A and B) at the 95% level (the same
test but for each sample group). Clearly show all the steps performed. State a
short conclusion and then an explanation of your conclusion relevant to the
context of this scenario.
2. For sample group B, how confident could you be in your conclusion? Explain
how you arrived at this result.
Helpful formula:
=average(cell range)
=stdev(cell range)
mean
sample standard deviation
=tdist(t test statistic value, df, number of tails)
this gives the area under the t-distribution in a tail(s), to obtain the p-value
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Module 4, Homework
Excel for p-values
Linear Regression, Part 1
Excel p-values (first use the table to estimate, then use Excel):
1.
2.
3.
4.
For a Type II hypothesis test with t*=1 and n=15 determine the p-value.
For a Type III hypothesis test with t*=1.8 and n=21 determine the p-value.
For a Type I hypothesis test with Z*=-2.16 determine the p-value.
For a Type III hypothesis test with Z*=-1.11 determine the p-value.
Linear Regression Problem:
The marketing manager of a large supermarket chain would like to determine the effect
of shelf space on sales of pet food. A random sample of 13 stores is selected and the
results are shown in the Excel data file PETFOOD.
a. Set up a scatter diagram to see if the potential exists for a linear relationship
between sales and shelf space.
Highlight from A1:B14
Click on the Chart Wizard icon
Select XY(Scatter)
Select next and get to the labeling window
Properly label the X and Y axis
Select Chart, Add Trendline
b. Perform an Excel Linear Regression analysis to determine the regression
coefficients.
Select Tools, Data Analysis, Regression
For the Y-Range, enter B1:B14
For the X-Range, enter A1:A14
Check the Labels box
In the new worksheet ply box type in: LR Output
c. Interpret the meaning of the slope.
d. Predict the average weekly sales of pet food for a store with 17 feet of shelf space.
e. Determine the error between actual sales and estimated sales for your answer in d.
f. Interpret the coefficient of determination for this situation.
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